General multivariable limits

For multivariable limits, the way my math books has taught me to prove they exist is to use the epsilon delta argument (for every epsilon > 0, there is a delta >0 ....). I have heard that for most cases you will almost never have to use this argument. Is this true? I know you can use the squeeze theorem on some cases but what about the others?

I doubt that your books taught you that as "the" way to prove limits exist. That is, of course, the definition and so is always introduced first. But all of the "properties" of limits that are true for single variable limits are true for multivariable limits. For example, if your function f(x,y,z) is a fraction U(x,y,z)/V(x,y,z), the limit as (x,y,z) goes to (a,b,c) of U if L and the limit as (x,y,z) goes to (a,b,c) is M, not equal to 0, then the limit of f as (x,y,z) goes to (a,b,c) is L/M.